Are the imaginary units of complex numbers anyhow related to vectors? I have studied introductory vector for use in physics ( introduction to vectors, products of vectors) that have vectors defined by unit vectors $ \hat{i} ,  \hat{j} , \hat{k} $
And i have also read about complex numbers of the form $a + ib$, which simply defines a point in plane.
I want to know are the two things related to each other, if yes, how?
 A: Complex numbers can be used in 2D geometry, but do not generalize to 3D. There is a generalization to 4D, called quaternions.
Vectors can be defined with any number of dimensions, and important cases are 2D and 3D. Algebra on vectors and on complex numbers partially overlap, but not much.
A: Actually there is a very cool embedding of the complex numbers into the vector space of $2 \times 2$ matrices over the real numbers. The mapping is as follows: $$a+bi \rightarrow aI + b \begin{bmatrix}0&-1 \\1&0\end{bmatrix}$$ with $I$ being the $2 \times 2$ identity matrix. The eigenvalues of the matrix representing the imaginary part are also $\pm i$ as well which has always seemed rather elegant to me. The two structures reflect each other naturally.
A: No. $\hat i=(1,0,0)$, $\hat j=(0,1,0)$, $\hat k=(0,0,1)$ and $i=(0,1)$ if defined as a point of complex plane.
A: No the two things are not related at all and notably $i\neq \hat i$.
Anyway in the compex plane the imaginary unit $i$ can be considered in correspondence with the unit vector $\hat j=(0,1)$ as any other complex number $$z=x+iy=(x,y)=x(1,0)+y(0,1)=x\hat i + y \hat j$$
